## Abstract

A celebrated unresolved conjecture of Erdős and Hajnal states that for every undirected graph H there exists ϵ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least n^{ϵ(H)}. The conjecture has a directed equivalent version stating that for every tournament H there exists ϵ(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^{ϵ(H)}. We say that a tournament is prime if it does not have nontrivial homogeneous sets. So far the conjecture was proved only for some specific families of prime tournaments (Berger et al., 2014; Choromanski, 2015 [3]) and tournaments constructed according to the so-called substitution procedure (Alon et al., 2001). In particular, recently the conjecture was proved for all five-vertex tournaments (Berger et al. 2014), but the question about the correctness of the conjecture for all six-vertex tournaments remained open. In this paper we prove that all but at most one six-vertex tournament satisfy the Erdős–Hajnal conjecture. That reduces the six-vertex case to a single tournament.

Original language | English |
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Pages (from-to) | 113-122 |

Number of pages | 10 |

Journal | European Journal of Combinatorics |

Volume | 75 |

DOIs | |

State | Published - Jan 2019 |

### Bibliographical note

Funding Information:The first author was partially supported by BSF grant 2006099 and ISF grant 1581/12. The third author was partially supported by NSF grants DMS-1550991 and BSF grant 2006099.

Funding Information:

The first author was partially supported by BSF grant 2006099 and ISF grant 1581/12 . The third author was partially supported by NSF grants DMS-1550991 and BSF grant 2006099 .

Publisher Copyright:

© 2018 Elsevier Ltd

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics